The dual probe heat pulse method: interaction between probes of finite radius and finite heat capacity

We analyze the dual-probe heat-pulse (DPHP) method for measuring the thermal properties of soil or other media. The method involves measuring the temperature rise of a receiver probe that is parallel to, and a known distance from, an emitter probe from which a pulse of heat is released. Under the assumption that the probes are perfect conductors, we derive a semi-analytical solution for this method that accounts for the finite radius and finite conductivity of the probes and contact resistance between probe and soil. The solution in the Laplace domain is obtained by writing solutions of the Helmholtz equation around each probe as infinite series of terms involving Bessel and trigonometric functions. Addition theorems are then used to write the solutions centred at each probe in terms of solutions centred at the other probe. Truncating the series and solving a system of linear equations gives numerical values for the series coefficients, which in turn gives values of the Laplace transforms for numerical inversion. We use the solution to investigate the validity of a simpler approximate solution that is being used in conjunction with the DPHP method for thermal property determination. For what we define as typical implementations of the method, our results show that error resulting from use of the approximate solution is sufficiently small that its effect on estimated thermal properties will be negligible. The same general approach can be used to investigate a growing number of DPHP applications for which the approximate solution may be of questionable accuracy.